An Approach to Planning and Control of
Stochastic Network Flows
by
Michael J. Paskowitz
Submitted to the Department of Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degrees of
Bachelor of Science in Computer Science and Engineering
and Master of Engineering in Electrical Engineering and Computer Science
at the Massachusetts Institute of Technology
,May 22, 2000
D 2000 Michael J Paskowitz. Altrights reserved.
The author hereby grants to M.I.T. permission to reproduce and
distribute publicly paper and electronic copies of this thesis
and to grant others the right to do so.
Author
Department of Electrical Engineering and Computer Science
May 22, 2000
Approved by
SteVhan E. 1i
The Charles Stark Draper Laboratory, Inc.
Technical Supervisor
Certified by
C'ynthia Barnhart
Codirector, MIT Operations Research Center
ThqKue risor
Accepted by.
(
Arthur C. Smith
Chairman , Department Committee on Graduate Theses
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An Approach to Planning and Control of
Stochastic Network Flows
by
Michael J. Paskowitz
Submitted to the
Department of Electrical Engineering and Computer Science
May 22, 2000
In Partial Fulfillment of the Requirements for the Degrees of
Bachelor of Science in Computer Science and Engineering
and Master of Engineering in Electrical Engineering and Computer Science
ABSTRACT
Large scale real-time planning and execution of complex flow-based networks requires
careful modeling and analysis. These real-time systems often need to be modeled for an
indefinite amount of time, and are often faced with unpredictable, stochastic changes to
the system over time. Real-time planning must be done progressively, adapting to change
as or after it occurs. Implemented plans must strive to be as efficient as possible while
conforming to network demand and resource constraints. This thesis presents an approach
to the planning and execution of stochastic real-time network flow problems. The
approach taken is to apply linear programming, a deterministic network solution
technique, to a stochastic real-time network flow problem with an unbounded time
horizon by solving finite sections of the entire problem, one by one, over time. The
approach is analyzed using a case study problem in the domain of military logistics
planning. The analyses offer insight into more general application of this approach to
other real-time network flow problems.
Technical Supervisor: Stephan E. Kolitz
The Charles Stark Draper Laboratory, Inc.
Thesis Supervisor: Cynthia Barnhart
Codirector, MIT Operations Research Center
2
Acknowledgements
There are many individuals to whom I owe my deepest thanks and gratitude for their
contributions to my thesis research.
First and foremost, I would like to thank Dr. Stephan Kolitz of Draper Laboratory and
Professor Cynthia Barnhart of MIT for their constant guidance and support. Even with
the many obligations of their own work, they both found the time and effort over the past
year to help drive my research to completion. I could not have come this far without their
collective experience, technical vision, and sage advice.
I am grateful to the Charles Stark Draper Laboratory for its commitment to education,
and for funding my MIT tuition.
Thanks also go to Bill, my Draper buddy, for his countless contributions to my work
experience.
My parents, Irv and Jean, deserve my highest praise for their faith and support over the
past year, and the 22 that preceded it.
Finally, now and forever, I would like to express my grateful appreciation and love to my
fianc6e Maria. Her motivation and encouragement have kept me working hard on this
project, and have shown me that there really is a light at the end of the tunnel.
This thesis was prepared at The Charles Stark Draper Laboratory, Inc, under Internal
Company Sponsored Research: Real-Time Large-Scale Optimization, IR&D Project
2036.
Publication of this thesis does not constitute approval by Draper or the Massachusetts
Institute of Technology of the findings or conclusions contained herein. It is published
solely for the exchange and stimulation of ideas.
Michael J. Paskowitz
22 May 2000
3
4
Table of Contents
1. INTR O D U CTIO N ................................................................................................................
7
2. INTEN T A ND O BJECTIVES.........................................................................................
16
2.1 THE PARTIAL PLANNING APPROACH..............................................................................
16
2.2 PARTIAL PLAN LENGTH EFFECTS..................................................................................
17
2.3 NETWORK PARAMETER FORECASTING AND ASSESSMENT..............................................
18
2.4 PLAN CONTINUITY .........................................................................................................
19
3. IMPLEM EN TA TIO N DETAILS...................................................................................
22
3.1 D EFINITIONS.................................................................................................................
22
3.2 FORM ULATING THE M ODEL ..........................................................................................
24
3.3 FINDING A SOLUTION.......................................................................................................
25
3.4 SOLVING A PARTIAL PLANNING SERIES ........................................................................
26
3.5 FLOW PIPELINING ............................................................................................................
28
4. THE CA SE STUD Y M ODEL .........................................................................................
30
4.1 TRANSPORTATION ........................................................................................................
31
4.2 CONTAINERS, DEPOTS AND PORTS................................................................................
31
4.3 N ET EXPLOSIVE W EIGHT ..............................................................................................
32
4.4 LATE D ELIVERY ............................................................................................................
33
4.5 D ECISION V ARIABLES ......................................................................................................
33
4.6 O BJECTIVE FUNCTION....................................................................................................
35
4.7 M ATHEM ATICAL FORM ULATION...................................................................................
36
5. A NA LYSES A N D RESULTS.........................................................................................
5.1 PARTIAL PLAN LENGTH ................................................................................................
39
39
5.1.1 PartialPlan Length Scenario Description.............................................................
40
5.1.2 D eterm inisticD emand Results................................................................................
42
5.1.3 Stochastic D emand Results........................................................................................
43
5.2 FORECASTING DEM AND................................................................................................
44
5.2.1 Forecastingand Assessment Scenario Description .................................................
45
5.2.2 ForecastingResults ...................................................................................................
46
5.3 TRANSPORTATION DELAYS..............................................................................................
5
48
5.3.1 TransportationDelay Scenario Description...........................................................
48
5.3.2 TransportationDelay Results .................................................................................
50
5.4 ON-DEM AND REPLANNING AND INTEGRATION .............................................................
51
5.4.1 On-demand Replanning Scenario Description.........................................................
52
5.4.2 TransportationLink Loss Event Results.................................................................
53
5.4.3 Depot and Port Loss Event Results.........................................................................
54
5.5 EXECUTION SIM ULATION..............................................................................................
56
5.5.1 Execution Sim ulation Scenario Description...........................................................
57
5.5.2 Sim ulated Execution Results...................................................................................
58
6. SU MMA RY A ND CON CLUSIO NS ..................................................................................
60
6.1 RESEARCH SUM M ARY....................................................................................................
60
6.2 CONCLUSIONS .............................................................................................................
62
7. FU TU RE W OR K ................................................................................................................
67
7.1 PARTIAL PLAN INTEGRATION TECHNIQUES...................................................................
67
7.2 PLAN CORRECTION M ETHODS ......................................................................................
68
7.3 APPLICATION TO OTHER PROBLEM S ................................................................................
69
APPEND IX .............................................................................................................................
71
SAM PLE AM PL M ODEL FILE ..............................................................................................
71
SAM PLE AM PL D ATA FILE ...............................................................................................
73
REFER EN CES .......................................................................................................................
75
6
1. Introduction
A flow-based network is a weighted, directed graph with two sets of specially
designated nodes, the sources and the sinks, and a capacity function that maps edges to
positive real numbers. The edges of the graph are generally directed from source to sink,
and the weights represent the maximum amount of flow that can travel along each edge
[2]. Flow refers to one or more commodities, either as discrete elements or continuous
streams, that are present at the sources and travel to the sinks through the various edges
of the network. Water flows from reservoirs to kitchen sinks by way of pipes and water
pressure tanks. Airlines transport millions of passengers a year between numerous cities
around the world using a limited number of planes. Distributors supply goods from
warehouses to customers through various distribution channels. Inherent in each of these
and many other flow-based networks is a problem that has to be solved: on which edges
should the commodities travel so that each sink receives its required amount of the
commodity in the most timely and efficient manner possible? Such problems are known
generally as minimum-cost network flow problems [ 1].
Parametersare values assigned to nodes and edges of a graph. For network flow
problems parameters include the amount of supply at each source, the flow capacity of
each edge, and the demand at each destination or sink.
Many networks also have
constraints specific to their particular problems. A constraint is a logical relation between
several variables, each taking a value in a given domain [3]. Constraints in a network
7
flow problem impose restrictions on flow beyond those stated by edge capacities. Values
associated with constraints on a network flow are also considered parameters of that
network. In the case of the airlines, each plane must have a flight crew and some number
of flight attendants, and must be fueled, cleaned, and loaded to some capacity before it
can take off. The overall schedule of flights is also subject to weather conditions, other
traffic at the airports, and maintenance requirements.
Certain types of network flow problems need only be solved rarely. Infrastructure
networks like water and electric utilities are determined when the network is built and
essentially fixed for a long period of time. A flow-based network with fixed edges and
edge weights (i.e., that do not change over time) is considered a static network [1]. In
designing a water supply network the problem is to determine the best way to supply
sufficient water to each destination while minimizing the total number of pipes used and
minimizing the costs of installation and maintenance. It might be twice as cheap to lay
pipes aboveground rather than bury them in the ground, but they might then require four
times as much maintenance. Once decisions like this have been made and the network is
in place, flow proceeds as intended and the problem is considered solved. When a link
fails, it is typically rebuilt to match its preexisting condition and maintain the same
network structure.
This approach works for some systems, but not all network flow problems are
static. A network flow problem that airlines must solve involves the routing of passengers
from city to city to meet their demand for air travel. The solution to this problem is a
8
flight schedule for the airline's fleet. In practice, such schedules face ever-changing
network conditions that must be taken into account. Planes can have mechanical
difficulties or fail to depart on time. Pilots and flight attendants may call in sick and need
to be replaced. An airport may close in the middle of a snowstorm.
Because the
conditions of an airline network change over time, any one solution to its scheduling
network flow problem is only viable for a short period of time. Thus the problem must
be re-solved regularly, and the flow of passengers redirected (to a different plane, or a
later time) to maintain a feasible solution. Flow-based networks with parameters that
change over time are called time-expanded or dynamic networks [1].
Another example of a dynamic network flow problem is supply-chain logistics
planning.
This problem is found in various forms in many commercial and military
situations. The network sources are supply warehouses of one or more goods. The sinks
are any destinations, perhaps retail stores, construction sites, or fielded forces, which
need a recurring supply of the goods stored at the warehouses.
The network flow
problem involves how to most efficiently transport goods from warehouses to
destinations with demand [1]. The military commonly deals with this type of problem.
In times of war, the military must supply its fighting forces with food, medical supplies,
and munitions.
This problem is dynamic since both supply network conditions and
demand requirements can change over time.
Small network flow problems can be solved either by inspection or some limited
amount of work. But a large network, like any of the transportation networks already
9
discussed, could have an extremely large number of possible solutions and hundreds of
thousands of decisions that need to be made. These decisions are complicated by perhaps
thousands of constraints that must all be considered and satisfied together. Any set of
decisions that satisfies all constraints on the network is afeasible solution to the network
flow problem. A feasible solution may often be found by heuristic guidelines, but it is
unlikely that heuristic methods will yield an optimal solution [6]. The optimal solution to
these problems is the one that both satisfies all the constraints and minimizes the costs
involved in executing the solution. Execution is the process of physically implementing
the theoretically planned solution and actually moving flow from sources to destinations.
Finding the optimal solution to a large network flow problem can be extremely
difficult, requiring a significant amount of computing resources and a great deal of time.
Established linear programming methods exist for formulating and optimally solving
these problems. The formulation involves setting down the decisions to be made in the
problem as collections of variables, defining parameters with single values, codifying
constraints as equations in those variables and values, and describing the objective as a
function to be minimized (or maximized). With such a formulation a computer can
manipulate the millions of resulting possibilities through proven rigorous methods and
produce an optimal solution [7].
Often these problems require integer values for some or all of their decisions.
UPS, for example, cannot fly 1.7 planes between Boston and Dallas. Mixed Integer
Programming is a specialization of linear programming in which some of the decision
10
variables must be integer. The general problem of solving a mixed integer programming
problem optimally is NP-complete [7]. Thus no polynomial-time algorithm for solving a
generic mixed integer problem has yet been found [5]. Typical real-world scheduling,
routing, and transportation problems are very large, and so are very expensive in time and
resources to solve as a whole. One approach to take is to decompose large problems into
smaller ones, and solve those directly [9].
The benefits to this approach are speed of
solution and economy of computing resources, but often the objective value suffers from
the artificial boundaries imposed on the problem. Applying a provably optimal algorithm
to the whole problem, rather than isolated pieces of it, is guaranteed to produce at least as
good a solution, and probably a better one.
For the dynamic flow networks discussed thus far, the dynamic parameters of the
network, those that change over time, can be either deterministic or stochastic in nature.
Deterministic parameters change over time according to some function, which can either
be known or unknown to network planners. If all other network parameters (including
time) are held constant, the deterministic network parameters will also remain constant.
Stochastic network parameters change according to random event incidence - their
values are determined by a probabilistic function. Even with all static and deterministic
parameters held constant, the actual realization of a stochastic network parameter cannot
be perfectly predicted. Many network flow problems that model real-world systems have
stochastic characteristics.
availability issues.
unpredictable demand.
Transportation networks face weather delays and personnel
Logistics and supply networks might have to accommodate
One way to model events in these systems is as stochastic
11
processes. The future generally cannot be known, only predicted. Network parameters
are expected to change, but the exact time and magnitude of such changes can only be
estimated.
Systems that model flow-based networks and include random events and
parameter values that affect the final solution are called stochastic network flows.
Solving stochastic network flow problems involves not only routing flow from source to
destination, but also reacting to events that randomly occur and adjusting the solution to
compensate.
Because stochastic network parameters change unpredictably, one way to attack
network flow problems involving stochastic parameters is to solve them repeatedly over
time.
The frequency of re-solution depends on the impact of random events on the
existing solution, which varies from problem to problem. Significant changes to network
parameters may require immediate replanning to avoid a drastic increase in total cost.
Stochastic network flows that result from airline flight plans and the execution of
logistics supply plans are real-time systems.
A real-time system in this context is a
dynamic network flow that models actual events and reacts to changes in network
parameters as (or soon after) they occur. These systems are often volatile, as unexpected
events can transpire with little or no warning, and require frequent replanning to maintain
feasible solutions. The time duration of many real-time planning systems is unbounded,
either because there is no desired end to the system being modeled or because the exact
time of such an end is unknown. UPS doesn't plan only to ship packages through next
12
week or next year, but indefinitely, with recurring (and possibly shifting) demand over
time.
It is undesirable to plan deterministically over even an approximation of an
unbounded time horizon because the conditions of the network will probably change. So
the question becomes, how can mathematical programming techniques be used in a
stochastic real-time system for planning and execution over an unbounded time horizon?
It is the goal of this thesis to explore an approach to planning and execution of
stochastic real-time network flow problems.
The approach taken is to apply linear
programming, a deterministic network solution technique, to a stochastic real-time
network flow problem with an unbounded time horizon by solving finite sections of the
entire problem, one by one, over time. The solution to a single finite section is called a
partialplan, because it plans the movement of flow for only part of the problem's time
horizon. We therefore refer to our approach as the partialplanning approach.
Figure 1 presents some common ways to classify network flow problems,
including static vs. dynamic networks, deterministic vs. stochastic events and parameters,
and fixed vs. unbounded planning time horizons.
We are interested in planning a
dynamic real-time system with stochastic network events, and will focus our analyses on
a military logistics planning problem.
This problem has the characteristics we are
interested in investigating, and a linear programming model is readily available for
adaptation [4].
13
Static Networks:
Water and Electricity supply networks
Dynamic Networks:
Deterministic
Network Parameters
Stochastic Network
Events
Fixed duration
planning horizon
Construction site
materials supply
Single season Cruise
Line ship schedule
Unbounded
planning horizon
Goods delivery network
to meet known and
recurring demand
Airline flight plans,
Military logistics
planning and execution
Figure 1: Classification of Common Network Flow Problems
The military logistics planning problem includes the movement of ammunition
from interior United States depots to coastal US ports. Stochastic events encountered in
this problem include transportation delays, the loss of network links between depots and
ports, and sudden shifts in demand if there is a crisis or contingency. Using this problem
as a case study, our goal is to gain insight into the best way to apply the partial planning
approach to real-time network flow problems under various network conditions.
In
addition, we shall examine the benefits of timely and accurate network parameter
estimation and the effects of stochastic events on plans already in place.
This introduction provides necessary background on the characteristics
of
network flow problems, and defines the focus of this thesis. Chapter 2 offers a more
14
detailed explanation of the objectives of the thesis. Chapter 3 explores the details of
solving
linear programs and running scenario
analyses,
and provides specific
implementation details for the solution methods used in this thesis. In Chapter 4, the
specifics of the military logistics planning problem are explained, and the problem
formulation is given.
Chapter 5.
Various scenario analyses and relevant results are presents in
Chapters 6 and 7 provide a summary of the research accomplished,
conclusions drawn from the analyses, and possibilities for future work.
15
2. Intent and Objectives
The intent of this thesis is to determine which methods for applying the partial
planning approach produce the least expensive solutions to various classifications of
network flow problems. In this chapter we define three objectives which will assist in
making that determination. Each objective involves the evaluation of various forms of
the partial planning approach as applied to specific network flow situations. We begin by
describing the details of the partial planning approach and the forms in which it can be
applied to various types of network flow problems.
2.1 The Partial Planning Approach
The partial planning approach to solving a network flow problem with an
unbounded time horizon involves the decomposition of the horizon into finite sections,
which are then solved using deterministic linear programming techniques.
The finite
solutions, referred to in Chapter 1 as partial plans, are executed serially over time to route
flow in the original, unbounded network flow problem.
Several aspects of the approach can be varied to produce more efficient, or costeffective, solutions to the same problem.
More efficient solutions are relatively less
expensive than other solutions. Variable aspects include the length of time for each
partial plan, the frequency of new partial plan generation, and the degree of continuity
16
between partial plans. Each aspect is described in greater detail in the following sections
of this chapter. Specifying a value for every one of these aspects defines a particular
variant of the approach. By applying a number of variants to the same network flow
problem, we will be able to determine the variant that produces the least expensive
solution.
The procedure for applying a specific variant is described in Section 3.4.
Within each variant, the values for partial plan length and replan delay time are consistent
- every partial plan is the same length, and each are combined with the same degree of
continuity.
Once a partial plan begins execution, the flow determined by that plan cannot be
changed. Events that occur during a partial plan's execution cannot be compensated for
by that plan. The effects of such events on network parameters have to be incorporated
into the generation of the next partial plan.
2.2 Partial Plan Length Effects
The first objective of this thesis is to determine the effects of the length of each
partial plan on the cost of a given network flow problem's solution. To isolate the effects
of plan length from those of the other aspects mentioned in Section 2.1, the variants of
the approach used for this determination all use a periodic replanning schedule and ignore
plan continuity. Periodicreplanning indicates that every partial plan finishes execution
before a new partial plan begins. Replans thus occur at regular time intervals over the
course of a problem's solution.
17
To fully appreciate the effect partial plan length has on the solution, we need to
consider two major types of network flow problems: deterministic and stochastic. We
expect that for deterministic networks, longer partial plan variants will produce
consistently lower cost solutions because each partial plan is able to optimize over a
longer period of time. In stochastic networks, however, the longer partial plan variants
will likely be adversely affected by each partial plan's inability to compensate for change.
Section 5.1 details the application of various partial plan lengths to both
deterministic and stochastic networks. The results found there provide insight into the
relationships between partial plan length and solution cost.
2.3 Network Parameter Forecasting and Assessment
Deterministic solution methods like linear programming depend on accurate and
timely information about the real values of network parameters. The termforecast refers
to a prediction of future network parameter values.
determining present network parameter values.
Assessment is the process of
We have already discussed how
stochastic events can affect the parameters of a flow network and cause the continued
execution of a currently executing partial plan to become extremely expensive.
A
forecast of the effects of such events on the network's parameters would allow a new
partial plan, which includes the new parameter information, to be generated and replace
the current one. (When a partial plan is replaced immediately in response to an event, we
18
say it is replanned on-demand.) If a forecast were unavailable, timely assessment of the
network's conditions after an event has occurred (and immediate replanning after the
assessment) would minimize the negative effects of the event.
Our second objective is to determine the qualitative value of timely event
forecasting and network parameter assessment.
We can make this determination by
examining how solution cost changes with the timeliness of a forecast (how long before
an event the event is predicted) or assessment (how long after the event its effects on the
network become known).
In actual real-time systems, accurate forecasts require
expensive predictive mechanisms that are based on current network parameters. Timely
assessments similarly require expensive constant network monitoring mechanisms. If a
real-time system already has such mechanisms in place, this objective should determine
the savings such mechanisms provide under a partial planning approach. Otherwise the
values derived for timely forecasts and assessments should indicate what savings could
be gained by implementing them.
Section 5.2 explores the values of forecasting and assessment for events that
cause sustained increases in a network's demand function.
2.4 Plan Continuity
An unavoidable result of using linear programming with the partial planning
approach is that the generated partial plans have no intrinsic relation to each other. Two
19
partial plans, whose network flow problems differ from one another only by a single
network parameter value, could be wildly different, but equivalent, solutions to almost
the same problem. The real-world context of network parameters cannot be transferred to
the linear program used to solve the problem.
Any information not formulated as a
constraint on the network itself is not even considered in the solution.
In practice, there are often situations in which other constraints, outside of those
in the linear program, need to be part of the solution. For the partial planning approach,
each partial plan is a coherent, optimal flow of supply from sources to sinks over the
length of the partial plan to meet demand. It is also desirable, however, to have coherent
flow from one partial plan to the next. We call this desire plan continuity - the need for
gradual change from one partial plan's flow schedule to the next. This is especially true
of systems that have significant human participation. Computers can adapt effortlessly to
plan discontinuity, possibly taking different actions to achieve the same result in each
successive partial plan. Human beings typically need time to adjust to such change,
especially if each partial plan establishes a pattern of flow that is then changed at every
partial plan boundary. During that time, called plan integration time, the new partial plan
is grafted onto the old one so that the change is reasonably gradual.
The mechanism through which two partial plans are grafted depends on the
specific problem being solved. It is beyond the scope of this thesis to precisely model
such a mechanism even for the case study problem, but we can approximate the effects of
plan integration time. During plan integration, the optimal solution cost of both the old
20
partial plan and the new one are sacrificed in order to allow the integration to occur. If
the new partial plan were generated in response to a network event the old plan was
incapable of handling inexpensively, then the most cost-effective action to take would be
to execute the new plan immediately. But if the two plans need to be integrated, then the
most cost-effective action cannot be carried out immediately. The new partial plan would
be executing in full only after the integration time, and would thereafter be providing
optimal flow. Thus the real effect of plan integration is a gradual shift from the solution
cost of the old partial plan to that of the new one.
We simulate this effect by leaving the old partial plan in execution during the plan
integration time, and then switching completely over to the new partial plan at the end of
that time. This results in slightly more expensive solutions overall, but provides a good
approximation of the effect of plan integration time on solution cost. We refer to our
approximation as replan delay time because the execution of new partial plans is delayed
rather than integrated.
Since many of the real-time systems to which we wish to apply our partial
planning approach require some degree of plan continuity, it is our third and final
objective to determine the effects of replan delay time on solution cost.
Section 5.4
specifically addresses the effects of replan delay time as applied to networks with
significant stochastic events. Section 5.5 also examines replan delay time, but does so
together with the other aspects of the partial planning approach in simulated stochastic
network execution.
21
3. Implementation Details
This section provides some details on the methodology of using linear
programming to solve network flow problems. The real problem must first be distilled
into collections of decisions to be made, and sets of parameters with values that model
actual network parameters. Once this has been done, the formulation can be codified and
used to generate results.
This section also defines the terms used throughout the
remainder of the thesis and describes the process of scenario analysis.
3.1 Definitions
Throughout this chapter and the remainder of the thesis we use the following
terms to describe certain aspects of the partial planning approach and the scenario
analyses presented in Chapter 5.
The term scenario is used to describe a specific real-world situation involving a
network flow problem that must be solved through planning and execution. An overall
planning horizon refers to the length of time over which an entire scenario occurs and
must be planned. Sections 5.1, 5.3, and 5.5 use long overall planning horizons to
approximate scenarios whose planning horizons would otherwise not be bound by a fixed
time duration. The overall planning horizon is decomposed into finite sections, and the
network flow problem in each section is solved to form a partial plan. When every
partial plan is put together to form a single plan for the overall planning horizon, that
22
A partial planning series
0-Overall
planning horizon .T
Analysis of a Single Scenario
1/3
03
ISolver
T
2/3 T'
Span of each partial plan
Overall plan solution value
0
Parameter that differentiates each series
Figure 2: The Anatomy of a Scenario Analysis
Each point on the graph is produced in similar fashion, by using a different parameter value and
solving a partial planning series. The scenario under analysis determines such values as the demand
function (shown here under the partial planning horizon) and other network parameters (not shown)
contained in the partial plans.
single plan is called the overall plan. The length of a partial plan is the number of time
periods over which it plans; the span is the set of time periods a partial plan contributes to
an overall plan. A (partialplanning) series is a collection of consecutive partial plans that
together make up an overall plan. Data used to analyze each scenario in Chapter 5 is
gathered by generating and solving a number of partial planning series, for the same
overall planning horizon, that differ only by one or two parameters. For each of the two
scenarios in Section 5.1, the parameter that varies from series to series is partial plan
length. The scenario in Section 5.5 contains series differentiated both by partial plan
23
length and replan delay time. The analysis then consists of plotting the gathered data and
interpreting the graph. Figure 2 illustrates the relationships between these terms.
The network flow problem contained in each scenario has a specific set of
network conditions.
A network's conditions are its parameters (like demand) and
characteristics (deterministic vs. stochastic demand, periodic vs. on-demand replanning,
etc.) We use the term stochastic network conditions to refer to a network with one or
more parameters or characteristics that behave unpredictably over time.
Stochastic
demand indicates that the demand function over a time horizon can deviate randomly
from the values used to plan for that horizon.
An overall solution is the cost of executing all partial plans in a series (in effect,
executing the overall plan for an overall planning horizon). This cost is the sum of the
execution costs expressed by each partial plan's objective function value. An efficient or
cost-effective solution is an overall plan for an overall planning horizon whose total
objective value (execution cost) is lower than that of other overall plans in the same
scenario analysis. An optimal solution is more efficient than any other solution to the
same scenario.
3.2 Formulating the Model
A problem formulated as a linear program and formatted for use with a
commercial solver has two basic parts: the model and the data. The model defines the
24
structure of the network (nodes and edges), the equation for the objective function, any
named sets (i.e. the set of airplanes or the set of supply warehouses), all parameters and
decision variables, and the equations (in terms of parameters and decision variables) for
constraints present in the network. In essence, the model represents an entire class of
possible problem instances. The data refers to all information that is specific to a single
instance of the problem defined in the model. It includes the members of defined sets
(i.e. the name of each warehouse) and the values for defined parameters. The demand
function and any equation coefficients are also declared as part of the data.
The
Appendix contains sample model and data files to illustrate the distinction.
3.3 Finding a Solution
Solving an instance of a problem requires feeding the model and the data to a
computer program designed to solve generic linear and mixed integer programming
problems. All problems presented in this thesis were solved using a commercial software
package, CPLEX 6.6.0 with AMPL. AMPL facilitates the formulation of the model and
data with its own programming language.
The model and data are first defined in
separate files, each with its own syntax. Then the two files are entered into the AMPL
command processor. The processor combines the model and the data into a complete
problem, formats the problem for solution, and sends the formatted problem to CPLEX.
CPLEX does the work of finding an optimal solution, and then passes the results back to
AMPL for display. The model and data files in the Appendix are written in AMPL's
programming language.
25
3.4 Solving a Partial Planning Series
Plugging a model and a data file into AMPL and finding a solution constitutes the
generation of a single partial plan. Since the model defines the structure of the problem
and not its specific instance, it remains largely unchanged for each partial plan of a single
partial planning series, except for minor modifications to the objective function.
Successive partial plans cover different spans of time from the problem's planning
horizon, so each plan requires its own data file. Thus a partial planning series for a long
overall planning horizon is typically made up of a single model file and a series of data
Planning Horizon Time
Demand Function
1
Partial Plan
Partial Plan 2
Partial Plan 3
Partial Plan 4
Data Files
Formatted
Problem
AMPL
1
-Results formatted
Model File
CPLEX
Optia
Result
for display
(Entered only once)
Figure 3: Solving a Partial Planning Series
Each data file contains a section of the demand function, as well as any other network
parameters. The model file is first entered into AMPL. Then a single data file is entered. AMPL
then sends the complete formulation to CPLEX, which solves the problem and returns the result
to AMPL. This process is repeated for each data file in the series.
26
files that contain separate sections of the network's parameter values (over time). To
solve a partial planning series, the model file is first entered into AMPL, and then each
data file is entered and solved in turn. See Figure 3 for a detailed illustration of this
process.
Each successive partial plan builds on the partial plans developed so far. The
partial plan for time periods 100-120, for example, is not solved in a vacuum; the plans
developed for periods 0-100 provide the initial network conditions (including flow) for
the 100-120 partial plan. These initial conditions are implemented by first loading the
solution to the previous partial plan into AMPL along with the new partial plan's data
file, and then fixing those loaded values. New partial plans take as fixed the decisions
made by all prior partial plans in a series, because in a real execution those decisions
would already have been carried out by the time the new partial plan takes effect. When
decision variable values are fixed, the solver treats them as network parameters.
Once the initial conditions are fixed, the objective function is changed to calculate
cost only for the span of time periods covered by the new partial plan. At the end of the
series, when all partial plans have been solved, the overall objective value of the solution
to the series is the sum of each partial plan's objective function value.
represents the cost of executing the overall plan generated by the series.
27
This value
3.5 Flow Pipelining
The overall plan produced by solving a partial planning series is used to schedule
flow through the network for the duration of the overall planning horizon. Regardless of
the length of each partial plan, their combination must manifest itself as a continuous
flow schedule over time. Even though an overall plan for 200 periods may have been
generated by a series of 20-period partial plans, the execution of that overall plan over
those 200 periods should seem as if the whole horizon were planned at once. If each
partial plan considered the end of its 20 periods as the end of the entire overall plan, flow
would stop temporarily at each boundary between consecutive partial plans. The partial
plans would have no reason to send any flow that would arrive after the ends of their own
spans.
In order to avoid stopping flow between spans of partial plans in a series, the flow
schedule must be viewed as a pipeline of flow from source to sink. The flow pipeline for
the case study network is made up of discrete trucks and trains. Each truck or train takes
a certain amount of time to travel from depot to port. Without flow pipelining, each
single partial plan sends only the necessary flow to fulfill demand for its span of time. If,
for example, a train takes four periods to reach its destination, no train would ever
intentionally be sent later than four periods before the end of a planning span.
Nevertheless, efficient flow across partial plan boundaries demands that the pipeline be
kept as full as possible. To this end, in every scenario in Chapter 5, each partial plan
schedules flow for several periods beyond its intended time span. When the next partial
28
Partial Planning Without Filled Pipeline
Plan 3
Plan 2
Plan I
No flow sent
after this time
Sink demand
met exactly
here. Pipe is
empty.
No flow arrives
next period. No
demand met until
Plan 2 schedules
new flow.
Plan 4
Similarly for these plans, no
demand will arrive in the
period after the boundary.
New plans must compensate.
Partial Planning With Pipeline Filling
Plan
10
Flow continue
to be sent upto
the boundary.
Plan 3
Plan 2
Plan 2 still takes over
at this point. But Plan
one has left the pipe
full. Flow will arrive
in this period.
Plan 4
Since previous plans scheduled
beyond their intended end, new
plans can expect some delivery
even if demand has changed,
easing plan transition.
Figure 4: Flow Pipelining across Partial Plan boundaries
Each Plan (shown as a horizontal line representing the time over which it plans) is a partial
plan in a series with the other plans adjacent to it. The dots represent the ends of each partial
plan's span. Note that with pipeline filling, partial plans actually extend beyond their spans.
plan takes over, the pipeline is not empty.
principle.
29
See Figure 4 for an illustration of this
4. The Case Study Model
The test problem for analyses in this thesis involves military logistics planning
and execution. The United States military has conventional ammunition storage depots
spread throughout the interior of the country.
During peacetime these depots act
primarily as long-term storage. But during armed conflicts, these depots supply the bulk
of US forces overseas with conventional munitions. Over the duration of the conflict, the
bulk of the munitions from these depots must be transported to ports on the proper coast,
and then taken by ship to the overseas theater of war. In a worst-case scenario, the
United States would have to sustain two full-scale armed conflicts, one in Asia (off the
West Coast) and one in the Near East (off the East Coast). Such a scenario would require
the use and supply of all interior depots and coastal ports, with sustained heavy demand.
This is the scenario assumed for analysis.
This chapter describes a network flow problem and a linear program formulated
to model the existing conditions of the military supply infrastructure. Munitions are
stored at five primary depots, which serve as the source nodes in the flow-based network.
There are three coastal ports, two on the West Coast and one on the East Coast. Because
we are modeling conflicts across both oceans, all three ports will be utilized. For the
purposes of this thesis, only the supply network from depot to port is considered.
Demand at each port is modeled as the recurring arrival of discrete ships, but the shipping
network itself is not a part of the test problem. Time is modeled in discrete periods; one
period is approximately equal to one day.
30
4.1 Transportation
Between the depots and the ports is a ground transportation network.
This
network involves two means of conveyance: truck and rail. Trucks typically have shorter
transport times than rail, but carry less capacity and are more expensive to operate. In
general, rail is the preferred mode of transport unless time constraints dictate the use of
more expedient means. For analyses involving stochastic transportation delays, rail
transport has a higher incidence of delay than truck transport [8].
4.2 Containers, Depots and Ports
The munitions themselves are modeled as the types of containers in which they
are shipped. Depots store munitions safely spread out in fortified bunkers.
Before
munitions can be shipped, they must be packed into shipping containers. The model
allows twenty distinct container types.
The container types represent different
configurations of munitions within a single container. Overall demand for munitions is
quantified as demand for each specific container configuration. Likewise, total supply at
each depot is composed of the supplies of each container type.
In our example problem, container stuffing at the depots takes one time period.
After being stuffed, the containers are assumed to be placed on staging areas in
preparation for shipping.
Each depot has limited stuffed-container storage space, of
which each type of container requires an equal amount.
Similarly at the ports of
embarkation, containers are unloaded and moved to limited-capacity storage pads for
31
Depot
00
Munitions Storage Facility
Stuffed Containers in
Staging Areas
-
Containers on Conveyances
En route to Destination Ports
Port
400
Queued Conveyances
awaiting unloading
Unloaded Containers on port
storage pads
Containers from port storage
pads being loaded onto ships
Figure 5: Munitions Movement through Depot and Port
eventual movement to and loading onto ships. It is from the container stock on these
storage pads that port demand is met. In addition, each depot has a container generation
rate that specifies how many total containers the depot can stuff per time period.
Likewise each port has a specific rate at which containers can be unloaded from
conveyances. Figure 5 illustrates the stages of container movement from depot storage
facilities to ship loading areas.
4.3 Net Explosive Weight
Safety is a primary concern when munitions are being moved. Restrictions on the
total effective amount of explosive material (Net Explosive Weight, or NEW) present at
any one place and time are imposed throughout the model. The container storage areas at
32
each depot and port are limited by NEW restrictions.
Each container type is assigned a
NEW value based on its contents; the number of each container type at a single location
multiplied by the NEW value for each container type determines the total NEW present at
that location.
4.4 Late Delivery
Although the ideal solution to the munitions transportation problem would fulfill
all demand requirements exactly on time, this goal is not always attainable. The model
allows for late arrival of demand, but assesses a heavy penalty for every period a
container arrives past its intended arrival time. Clearly late arrivals are discouraged, but
sometimes necessary if demand increases unexpectedly during plan execution or if the
system does not have sufficient throughput capability to meet demand.
Late flow is
modeled by network edges that progress backwards through time, one period at a time, at
port storage pads. Thus a container that arrives at time 3, but was intended to arrive at
time 1, would flow on late arcs from time 3 to time 2, and then from time 2 to time 1. In
traversing two late arcs, the container would thus be penalized for arriving two periods
late.
4.5 Decision Variables
The decisions to be made in the course of generating a feasible plan to solve this
problem are defined as decision variables in the model. Figure 6 illustrates the flow of
33
Time
0
1
2
3
4
5
6
Containers being stuffed at depot
W0\
Containers
awaiting
loading
R
_0
Conveyances in transit
Containers on
conveyances
queued at port
Containers
transferred to
storage pads
X
Q
P
Containers waiting for ships
L
Late arrivals
intended to
Containers being loaded onto ship (satisfying demand)
satisfy past demand
Figure 6: Container Movement Network, from Depot to Port over Time
Each edge represents a decision variable in the model. Each row of nodes is a single physical location.
Each column is the flow to and from every location in a single time period. Arcs that travel vertically carry
flow from one location to another. Horizontal arcs represent flow that remains in the same location for
more than one time period. The bold edges illustrate the flow of a container stuffed at time 2 and intended
for delivery at time 3. Although containers don't actually travel back in time, they flow to earlier time
periods to fill unsatisfied past demand and indicate the number of late containers per period.
munitions from depot storage facilities to port shipyards in a time-space network [1].
Each node in the figure represents a physical storage location at a depot or port, and each
edge represents the movement of munitions. The quantity of ammunition associated with
each edge is determined by a single decision variable value. Time proceeds from left to
34
right; each horizontal row of nodes is actually the same location at a different time. The
edge labels correspond to the decision variables as described in the model formulation
(Section 4.7), and text labels are also provided. For the sake of simplicity, Figure 6 only
depicts a single depot and port, and edges for the flow of one container type. The
complete problem network has similar connections to the one shown, but many more
nodes and edges.
4.6 Objective Function
The best solution to this problem delivers all required demand in the most costeffective manner possible.
Several factors contribute to the total cost of munitions
shipped during a plan's execution. Each truck or train costs some amount per container
to operate.
A severe penalty for late containers is applied in order to ensure timely
delivery whenever possible. Transporting any amount of explosives is a dangerous
undertaking, and transporting more at once increases the chances of an accident
occurring; therefore a penalty for the total NEW in transit per time period is assessed to
limit the total amount being transported at any one time. Finally, there is an added cost
for queued conveyances at ports; while they are waiting to be unloaded, they are not
being used for further transit. The exact function, in terms of decision variables and
network parameters, is given in Section 4.7.
35
4.7 Mathematical Formulation
The mathematical formulation for the test problem, taken from [4], is given
below.
The linear program used to generate the results found in this thesis is based
directly on this formulation.
SETS
* Ns
* Nd
"
"
V
B
-
The
The
The
The
set
set
set
set
of depots
of ports
of conveyance types
of container types
DECISION VARIABLES
* Xbijkt
- the flow of container type b e B from depot i E Ns to port
conveyance k e V at time t = 0, 1, 2, ...
j
e Nd on
jkt
-
the number of containers of type b e B on conveyances of type k e V
enqueued at port j E Nd at time t = 0, 1, 2,...
" Ybjkt
-
the number of containers of type b e B at port j E Nd that are unloaded
from conveyances of type k e V at time t = 0, 1, 2, ... and taken to port
storage pads
*
-
the number of containers of type b e B at port j e Nd that are located on
port storage pads and ready for retrieval for ship loading at time t = 0, 1,
" Q
Pbjt
2, ...
" Rbit
-
the number of containers of type b e B already stuffed and waiting in
staging area for transportation at depot i e Ns at time t = 0, 1, 2, ...
" Wbit
-
the number of containers of type b e B stuffed and delivered to staging
pads for transportation at depot i e Ns at time t = 0, 1, 2, ...
*
Ljb(t+1)t
-
The number of late containers of type b e B which arrive one period late
(from time t+1) at time t = 0, 1, 2, ... at port j e Nd.
36
PARAMETERS
"
Tijk
-
travel time from depot i e Ns to port j e Nd on conveyance k e V
"
sb
-
supply of container type b e B at depot i e Ns over the entire planning
horizon
" d"jt
-
demand of container type b e B at port j e Nd at time t = 0, 1, 2,
"
c ijk
-
unit cost of moving container type b e B from depot i e Ns to port
Nd on conveyance k e V
"
gjk
-
maximum number of containers unloaded from conveyance k e V at port
j e Nd per time period
*
nb
-
Net Explosive Weight of container type b e B
Sli
-
maximum accumulation of NEW allowed on storage pads at port j e Nd
fjk
-
detention costs per container per time period on conveyance k e V
queued at port j e Nd
-
the maximum container stuffing rate for depot i e Ns
STli
-
maximum accumulation of NEW allowed in staging areas at depot i e Ns
g
-
unit safety penalty cost of NEW in the transportation network
llcjb
-
the per-period cost of late arrival of container type b E B to port j e Nd
prijb
-
the relative priority of on-time delivery of container type b e B to port
E Nd
*
*
*
OBJECTIVE FUNCTION
Minimize:
cX
t=O
be B
(ij,k)
+
f
je Nd keV
Q
beB
n
+g
beB
T
+~ I
t=O
I
I1c
jb
pri
jb(t+L)t
jENd beB
37
R
iE Ns
Q ,+p
+
je Nd
kEV
j
e
j
CONSTRAINTS
Subject to:
T
Vi e Ns,be B
t=b
W b+R
_
-R,
b
=0
-XXt
Vie Ns,be B,t=0,...,T
j,k
Xb
+Q
(t1-Q
Yb =0
Vje Nd,k eV,be B,t =0,...,T
ieNs
Jyb
+pb
Yjk(t-1) +Pa
j(t-1)
pb
it
dbjt
Vj e Nd,be B,t =0,...,T
keV
Vie Ns,t=0,...,T
beB
Vje Nd,k eV,be B,t =0,...,T
beB
nR
Vie Ns,t=0,...,T
bEB
Vj e Nd,t =0,...,T
bE=B
All Al
Flows W
Rbit' Xhijkt'jQkt'I
Qb Yjkt' pb
L
Fow
Wt
jtIb(t+1)_
>0 and integer.
38
5. Analyses and Results
For an explanation of the terms used throughout this chapter, refer to Chapter 2 and
Section 3.1.
In this chapter we present a number of scenarios that model situations
encountered during the planning and execution of real-time network flows.
Each
scenario is planned using specific variants of the partial planning approach. These
variants of the approach include altering the length of each partial plan, generating new
partial plans on either a periodic or on-demand replanning schedule, and changing the
allotted replan delay time for partial plan integration.
Each analysis is designed to
provide insight into the effectiveness of the partial planning approach as it could be
applied to a real-would scenario. The analysis of each scenario consists of solving a
number of partial planning series (as described in Section 3.4), each of which uses a
different variant of the partial planning approach, and determining which variant
produces the most cost-effective solution to the network flow problem in that scenario.
5.1 Partial Plan Length
The munitions logistics planning problem involves the transportation of munitions
from interior depots to coastal ports to satisfy the demand for munitions overseas. This
demand comes from US involvement in a conflict overseas, and must be sustained for the
39
duration of involvement. The duration of the conflict, and consequently the duration of
demand, is not known but assumed to be extensive. A transportation schedule needs to be
devised to move ammunition efficiently from depots to ports through a ground
transportation network for the duration of the conflict. Once devised, the schedule must
be executed and munitions shipped according to it. Since the extent of the planning
horizon for this problem cannot be determined, a sensible approach is to solve and
execute successive partial plans, each spanning a finite length of time, until supply is no
longer demanded.
5.1.1 PartialPlan Length Scenario Description
At the start of the conflict, the transportation network is assumed to be empty - all
conventional munitions are in depot storage facilities. Once the demand for munitions
begins, a partial plan is solved for some number of periods, and depots start to send
supply to ports following that partial plan. When the partial plan ends, a new partial plan
is solved and executed. In real-world execution this process would continue indefinitely,
or until demand ceases. In this scenario, indefinite demand is approximated using an
overall planning horizon significantly longer than the length of a single partial plan. The
analysis is concerned with optimizing the solution to a long overall planning horizon
using a periodic replanning approach. For the purposes of this analysis, replans only
occur periodically, and each partial plan in the same overall plan has the same length.
We are also interested in discovering how overall solutions generated using the periodic
replanning approach differ depending on deterministic or stochastic network conditions.
40
We present two separate scenarios for comparison. The first contains a strictly
deterministic demand function that changes predictably over time, and no stochastic
network events. The second incorporates stochastic demand changes and transportation
delays. In both scenarios, once a partial plan begins execution, it cannot be changed - the
plan must be followed exactly. To analyze each scenario, we define a single demand
function over a 400 period horizon. We then use the process described in Section 3.4 to
solve a number of partial planning series. Each series is solved using a different length
for the partial plans. When the objective function values for each series' solution is
plotted against the partial plan length for that series, the relationship between partial plan
length and overall objective value for the particular network conditions of the scenario
will be known. These relationships satisfy the first objective of this thesis (see Section
2.2).
In the stochastic demand scenario, several types of events could occur over the
course of a single partial plan's execution. First, demand for munitions could change
unexpectedly. Second, the truck and rail transportation plans produced by each partial
plan may not be met during execution. In the model transportation delays occur with
small probability each time a vehicle is sent from a depot to a port. For rail, the values
used are: 5% chance of single-period late arrival, 2% chance of three-period late arrival,
and 1% chance of five-period late arrival. All trucks have a 4% chance of being delayed
one period. These numbers are not derived from actual data, but are intended to produce
sufficient delay to affect solution quality without rendering all partial planning series
solutions extremely inefficient.
41
In both scenarios the demand profiles (at the port of embarkation) are derived
from a schedule of ship arrivals and departures. Each port has a schedule of ships that
arrive to be loaded, depart for abroad to be unloaded, and return to be loaded again. In
the stochastic demand scenario, increases in munitions requirements overseas cause the
temporary addition of extra cargo ships to each port at various times during the scenario's
execution.
5.1.2 DeterministicDemand Results
Figure 7 shows the relationship between partial plan length and overall objective
value in the deterministic demand scenario.
In the absence of stochastic events, the
Partial Plan Analysis with Deterministic Demand
1400000
4)
1200000
C 1000000
0
800000
600000
i 400000
R nw
200000
0
0
50
100
150
200
Partial Plan Length
Figure 7: Periodic Partial Planning with Deterministic Demand
(Lower overall solution value is better)
42
250
model can accurately predict and schedule on-time delivery of all demanded munitions.
Long partial plans achieve greater solution efficiency and drive down the objective cost
of plan execution because they each have more information with which to make optimal
decisions.
5.1.3 Stochastic Demand Results
Figure 8 relates partial plan length to objective value when the transportation
network and demand function are subject to stochastic events. The results show that the
events have a negative impact on both small and large partial plan lengths. Very small
partial plans suffer because they may need to send large amounts of ammunition to
Partial Plan Analysis with Stochastic Demand
45000000
40000000
35000000
30000000
25000000
-
0> 20000000
15000000
0U U U
-
i10000000
0 fAfI f
0
.
-
........ ------*......
.
5000000
0
-
0
.....
.
.. -------6.......
-
20
40
60
80
100
Partial Plan Length
Figure 8: Periodic Partial Planning with Stochastic Demand
Also includes random transportation delays. Lower overall solution value is better
43
120
supply excess demand from previous events, and lack the time to do so at low cost. Very
long partial plans also perform poorly because events that occur early in each partial
plan's execution are not compensated for until the beginning of the next partial plan,
causing significant late demand penalties. In-between these two extremes a better
solution can be obtained. Mid-length partial plans (about 40 periods long) are able to
recover faster than long partial plans from stochastic network events, and send flow more
efficiently over longer periods of time than short partial plans.
5.2 Forecasting Demand
Warfare is inherently unpredictable. Logistics support must be able to adjust to fit
the needs of the battlefield. Without knowing the future, logisticians can only act based
on prediction and intelligence.
In a real-time system, information about what has
happened before and what might happen soon has a serious impact on the immediate
decisions that must be made. Better forecasting and information assessment leads to
overall plans that more closely match actual conditions and result in more efficient
solutions.
Our second objective is to answer the following question: what is the qualitative
value of more accurate information on past, present and future demand conditions?
Accurate information is essential to achieving optimality when applying deterministic
planning methods to scenarios in which substantial stochastic demand events can occur,
but such information is not always available. We examine this question by focusing on a
44
single, sustained increase in overall demand. Although several such increases can occur
over the course of a munitions supply scenario, we assume they are spaced by a
reasonable amount of time and so can be considered independently.
Between spaced
demand increases, the system should adapt itself to providing the new level of demand
and return to a steady-state transportation schedule.
5.2.1 Forecastingand Assessment Scenario Description
The scenario for analyzing the value of forecasted information consists of a single
increase in demand that occurs within a period of time covered by two consecutive partial
plans. Other than this single demand increase, the scenario contains no stochastic events.
The first partial plan, implemented sometime before the change in demand was
discovered, is developed with no knowledge of the change.
As soon as the event
becomes known, a new partial plan is solved on-demand and executed with perfect
knowledge of the event's effects on the demand function. The event becomes known
either through forecasting (before it actually occurs) or through an assessment of current
network conditions (after the event occurs). It is plausible for stochastic network events
that occur in a real-world scenario to remain unknown for several periods after the
occurrence. In this analysis, the first partial plan of the series begins at time 0. Its end
(and the second partial plan's beginning) is varied from period 20 to period 40. The
crossover point between the first and second plans marks the time when the event
becomes known. The second plan always ends at period 60. The increase in demand
always occurs in the middle of the scenario, at time 30.
45
An accurate forecast of the event before it occurs would permit the planning and
execution of a new partial plan whose solution is already prepared to handle the effects of
the event when it actually does occur. Qualitatively worse forecasts and assessments,
those that only perceive the shift in demand at or after the time it occurs, will have an
increasingly negative effect on the solution to the overall plan. By varying the time at
which the first partial plan ends and the second begins, we should see a progressive
worsening of the overall solution as the end of the first partial plan approaches and then
extends past the increase in demand. The second partial plan has increasing execution
cost in order to supply the excess demand as quickly as possible.
5.2.2 ForecastingResults
The results of this scenario analysis for two different increases in demand are
given in Figures 9 and 10. The first scenario involves a moderate increase in demand; the
Moderate Demand Increase at Time 30
8 5000000 -
> 4000000
-
3000000
4 2000000
1000000
0
-
0
20
25
30
35
Time Period dividing Plan 1 and Plan 2
Figure 9: Forecasting of Moderate Demand Increase
(Lower overall solution value is better)
46
40
second scenario's event nearly doubles the previous level of demand.
In both scenarios,
when the first partial plan ends between periods 20 and 25, the overall solution does not
suffer significantly from the demand increase because the second partial plan has ample
time to ship the extra required supply. As the forecast gets worse and the second partial
plan begins execution closer to the occurrence of the event, the effect on the solution
becomes gradually more important.
As the first partial plan begins to remain in effect after the event occurs, the
solution is sharply affected. If the assessment of network conditions is extremely late, the
event is not known until well after it occurs and the second plan doesn't begin execution
until a great deal of demand has failed to be supplied on time. In this case the objective
continues to increase steadily with the number of periods between the shift and the
Sharp Demand Increase at Time 30
S10000000
>C
8000000
0
6000000
S2000000 O
NO
00
20
25
30
35
Time Period dividing Plan 1 and Plan 2
Figure 10: Forecasting of Sharp Demand Increase
(Lower overall solution value is better)
47
40
beginning of the second partial plan. Once the new partial plan finally takes effect, it
sends as much supply as possible to make up for the late demand as quickly as it can.
The greater cost of faster transportation therefore also increases the overall solution
value.
5.3 Transportation Delays
The transportation network for the munitions logistics planning problem is made
up of truck and rail links from each depot to each port. The specifics of highway routes
and train stations are not important to this analysis, but the events that can transpire while
vehicles are in transit are very important. Trucks can get a flat tire or break down, and
take a day or more to repair. Train schedules can sometimes experience several days'
worth of delays. This analysis seeks to determine the effects of varying the incidence of
stochastic transportation delay events on long planning horizons solved using the partial
planning approach.
5.3.1 TransportationDelay Scenario Description
In the scenario analyzed here, a long time horizon of deterministic demand is
decomposed into periodic partial plans similar to those in Section 5.1. The first partial
plan is then solved, generating a transportation schedule. Before the next partial plan is
executed, that schedule is modified stochastically to reflect delayed truck and train
arrivals. Specifically, the decision variables that determine when trucks or trains leave
the depots (variable X from Section 4.7) are altered such that certain vehicles leave later
48
than originally scheduled. The delays range from 1 to 10 periods, and are determined
probabilistically for every shipment that traverses the network.
Trains are given a
slightly higher rate of delay incidence than trucks. The effects of these schedule changes
are identical to what would happen if, during real overall plan execution, a vehicle were
delayed on the road or track. Once these changes have been made, the transportation
schedule decision variables are fixed. The next partial plan must compensate for the
fixed late arrivals by assigning flow on the model's late arcs. Late flow is then penalized
by the objective function.
Two partial planning approach variants are applied to the scenario described
above in this analysis, each with a different partial plan length. Each partial planning
series is run with a different level of stochastic activity (i.e. with different percentage
chance of event incidence).
A few events over the course of a series should have
20 Period Periodic Partial Planning
9000000
8500000
*
8000000
>
7500000
7000000
6500000
U)
6000000 5500000
>
5000000
-
4500000 -.
-
-
.--
4000000
0
5
10
15
% Delays in Transportation Network
Figure 11: Transportation delay effects on a 20-period rolling horizon
(Lower overall solution value is better)
49
relatively minor impact on the overall objective value. As the percentage of late arrivals
increases, the efficiency of the plan should decrease rapidly. By comparing different
horizon lengths with various levels of transportation delay, we should gain some
understanding of the relationship between the partial planning methodology and the realworld risk of short-term network link delay.
5.3.2 TransportationDelay Results
Figures 11 and 12 depict the results of this scenario analysis on 20- and 50-period
partial planning approaches. Both show the increasingly negative effect of delays on plan
efficiency.
Once a plan partial is set in motion, its schedule must be followed.
Unexpected delays cannot be compensated for until the next plan, when supplemental
supply can be sent to cover long delays and incoming shipments can be redistributed to
50 Period Periodic Partial Planning
8000000
0 7000000
6000000
0C
5000000
2 4000000
) 3000000
L
2000000
0 1000000
0
0
5
10
15
20
25
%Delays in Transportation Network
Figure 12: Transportation delay effects on a 50-period rolling horizon
(Lower Overall Solution Value is better)
50
amortize the cost of extremely late deliveries. At low levels of stochastic transportation
activity, the longer 50-period partial plans are significantly better than the shorter 20
period plans. This can be explained by the benefits of having more information in each
partial plan and a longer span of time over which to optimize. As the results in Section
5.1.2 indicate, longer plans can achieve greater solution efficiency for deterministic
demand. Increased levels of stochastic delay take a greater toll on the overall solution
efficiency of longer partial plans, but the effect is not sufficient to entirely cancel out the
benefits of longer-term planning.
5.4 On-Demand Replanning and Integration
Sections 5.1 and 5.3 focused on variants of the partial planning approach that
allow only periodic replanning. We now turn our attention to on-demand replanning. Ondemand replanning would be extremely useful in scenarios where significant stochastic
network events could occur. A train wreck that destroys a shipment of ammunition and
renders the track useless is a good example of a significant stochastic network event.
During a real plan execution, such an event would prompt an immediate replan of part or
the entire network to adjust for the loss. In the case of the train wreck, the lost shipment
would have to be re-sent and all flow rerouted to avoid the disabled rail link.
The
drawback to on-demand replanning is that newly generated partial plans must be
integrated with the currently executing partial plan (see Section 2.4).
51
Significant stochastic events can be modeled in much the same way as minor
events were in Section 5.3: by modifying existing schedules between replans.
simulate lost munitions, transportation decision variables
To
are set to zero and
corresponding demands are left unsatisfied. The effects of network link failures can be
pushed forward through time by eliminating all scheduled flow on certain depot-port
connections. Once these changes are in place, the new partial plan would then take as
fixed the modified schedule (because by the time the new partial plan takes effect, the
fixed time periods will be in the actual past) and produce a new solution to the problem.
The new solution both recovers from the event (by making up lost shipments) and plans
delivery for the span of the new partial plan.
5.4.1 On-demand Replanning Scenario Description
The analysis in this section plots the quantitative impact of certain network events
against the required replan delay time as discussed in the third objective (Section 2.4) of
this thesis. Replan delay time is the time that passes between when a replan is generated
and when it begins execution. If a new partial plan is generated when a significant event
occurs, and it can take effect in the next time period, the impact should be minimal. All
that would be required is to reroute flow around failed links and reship lost cargo. The
longer the old partial plan remains in effect after the event occurs, the greater the negative
effect of the event on the efficiency of the overall solution. The new partial plan will
have to trade off higher costs to recover as quickly as possible from the past event.
52
The scenario analysis in this section is similar to that of Section 5.2, with a 60period horizon broken into two plans and a network event occurring at period 30. The
first plan is always unaware of the event, and the second always has perfect knowledge of
it. The difference here lies in the types of events and their effects on the network.
Demand is now deterministic for each scenario.
Although each event does not occur
until period 30, the set of series within each scenario begins with the first partial plan
ending at period 25. The overall solution values for series in which the first partial plan
ends before the event occurs are intended to indicate the effect of the event if it were to
be predicted and planned for before it happened.
5.4.2 TransportationLink Loss Event Results
Figure 13 compares the solutions for several realistic scenarios, including both the
partial and total loss of truck or rail links for a fixed amount of time. Events such as train
wrecks, rail line closings, major highway construction, and maintenance at depot or port
facilities could trigger these network effects. In all cases, integrating a new partial plan
within four periods after the event occurs can minimize the effects of link loss. The few
periods of lost delivery can be made up with excess link capacity elsewhere in the
network in the first few periods of the new plan.
If the new plan takes longer to
implement, there will be more demand to deliver than the new plan can easily handle. As
the sum of past lost demand (due to the event) and current demand requirements exceeds
total per-period link capacity, the network becomes saturated and the cost of substantially
late deliveries dominates the objective function. As the figure shows, the loss of rail
53
Replanning after a Significant Event at time 30
900000 800000
8 700000
> 600000
00 500000
400000
300000
o
200000
100000
0
25
27
29
31
33
35
37
39
Replan Crossover Period
-+-
No trucks to Concord (15) ---
-A-
Total Rail Loss (10)
No rail to Port Mots u (15)
-a- Total Truck Loss (10)
Figure 13: Significant Event Replanning.
Crossover Period is the time period when the replan takes effect. Concord one of two ports on the West
Coast. Motsu is the only port on the East Coast. The number in parenthesis is the duration of the event.
transport has a significantly greater effect on the overall solution than the loss of truck
transport.
5.4.3 Depot and PortLoss Event Results
Events such as inclement weather conditions at depot loading or port unloading
facilities, exhausted supply, or major facility maintenance would manifest themselves as
the loss of a depot or port for some length of time during plan execution. In the case of a
lost depot, no supply can be sent from that depot for the duration of the outage. In the
case of a port, no shipments can be made to that port from any depot. In addition, the
54
demand at that port is shifted to another port. This would be analogous to rerouting
inbound ships that had been destined for the closed port.
The effects of this type of event are shown in Figure 14. The loss of a single
depot is covered by increased output at the other depots. The effect on overall objective
value is not particularly significant unless the replan occurs far past the incidence of the
event. Each depot shares the supply burden reasonably equally, so when they are all
operating none is at maximum supply capacity.
When one depot stops supplying
ammunition, the rest are able to pick up the extra burden reasonably efficiently. As the
Replanning after a Significant Event at time 30
2500000
-
2000000
0
1500000
1000000
C$1
100000
5
0
25
27
29
31
33
35
37
39
Replan Crossover Period
-*-1 Less Depot -a-1 Less Port
Figure 14: Significant Event Replanning
Crossover Period is the time period when the replan takes effect. The loss of a depot lasts from time 30 to
the end of the scenario (at time 60). The loss of a port lasts for 20 periods.
55
burden increases (with later replanning), the other depots incur the greater cost of quicker
transportation via trucks in order to avoid the still greater cost of significant late delivery.
The loss of a port has an immediate and serious impact on the network. In this
scenario, one of the West Coast ports, Hadlock, was removed from the network for 20
periods.
During that time, the other port on the same coast (Concord) assumed the
schedule of ships intended for Hadlock.
All truck and rail shipments destined for
Hadlock in the original plan were treated as lost. In a real scenario those shipments
would be recovered and rerouted to a new destination, but here we assume that no intransit rerouting is possible within the time frame of the planning horizon. Since the
original partial plan met but did not exceed Concord's usual demand schedule, the effect
on the objective function of increased demand at Concord is immediate. Even before the
event occurs, the second partial plan increases rapid delivery to Concord in anticipation
of the excess demand. The value of forecasting (as discussed in Section 5.2) is also
apparent in this analysis, as prior knowledge of Hadlock's closing seems to be the only
way to maintain a relatively efficient solution.
5.5 Execution Simulation
The final scenario analysis of this thesis seeks to combine the individual elements
explored thus far into a more realistic picture of long-term real-time planning and
execution for the munitions logistics planning problem.
56
5.5.1 Execution Simulation Scenario Description
For the following scenario, variants of the partial planning approach with ondemand replanning are applied to a long planning horizon. During the course of plan
execution, stochastic events such as single-shipment transportation delays, network link
losses, and short-term demand increases can occur. We can gain insight into the most
effective variant of the approach by solving several partial planning series with different
partial plan lengths and replanning delays, and plotting their overall objective values for
the entire planning horizon. The probability of stochastic event incidence is relatively
low (around 5% per event) and remains constant across each series.
The series solved in this analysis are actually run as simulated planning and
execution of partial plans over a 400 period planning horizon. At the beginning, a partial
plan is solved for a deterministic demand function. At each period during the execution
several events can occur with various small probabilities.
These events are single rail
link outages (for 10 periods), demand increases (for 20 periods), and the loss of a port
(for 30 periods).
Each of these major network events is temporary, and triggers an
immediate replan. The time between the event and the institution of the new partial plan
varies from series to series. If no event occurs before the currently executing partial plan
reaches the end of its span, a new partial plan is solved and begins executing
immediately. Individual transportation delays (as in Section 5.3) occur at random
throughout the entirety of the scenario's execution, but do not prompt replans. At the
start of each new partial plan, the transportation schedule for all prior time periods is
fixed. The new plan must account for any missed demand that may have occurred in the
57
past, as well as any other events that occurred during the execution of the previous partial
plan.
5.5.2 Simulated Execution Results
Figure 15 plots simulated series solutions for three different partial plan lengths
and three different replan delay times. Longer partial plans are more cost-effective than
shorter ones, even in the presence of stochastic network events, because with on-demand
replanning they are no longer severely penalized by the effects of major events early in a
On-Demand Replanning Scenario Analysis
10000000
-
9000000
8000000
W
_
7000000
6000000
0
.
5000000
0
CD
4000000
0>
3000000
2000000
1000000
0
I
0
1
2
3
4
5
6
7
8
Replan delay time
-+-
30-period -m- 50-period
-&-
80-period
Figure 15: Simulated Overall Plan Execution with On-demand Replanning
Each of the three data series in this figure represents the planning and execution of the same scenario
using different partial plan lengths, as indicated by the label for each data series.
58
partial plan's execution. These results agree with the results from Section 5.3. With ondemand replanning to minimize the effects of other network events, transportation delays
are a major factor in the overall solution value.
Each partial plan, even if preempted
before finishing execution by a network event, still schedules transportation for
maximum cost efficiency over its entire partial plan length. Thus the periods between the
start of each partial plan and the event that ends it are served more efficiently by longer
partial plans. Additionally, the incidence of major network events is sufficiently small
enough that more of the shorter partial plans will finish execution and require periodic
replanning than 50- or 80-period plans. The same inefficiencies discussed in Section 5.1
also affect the shorter partial plan's periodic replans in this scenario.
Replan delay time has considerable negative effect on the overall solutions,
especially for shorter partial plans. This effect is similar to the one studied in Section 5.4.
With little or no delay between event occurrence and replanning, all scenarios retain a
similar objective value. The solutions are only differentiated by the effects of partial plan
length and transportation delay.
As replan delay time increases, its effect becomes
significantly more pronounced in the shortest partial plan approach.
Already less
efficient partial plans are left in place longer, compounding the negative effects. The
longer partial plan approaches are similarly affected by longer replan delay times, but
with more efficient flow within each partial plan, the effect is not nearly as pronounced.
59
6. Summary and Conclusions
The purpose of this thesis is to explore an approach to solving real-time stochastic
network flow problems. The important characteristics of these problems that must be
considered are stochastic network parameters and events, unbounded planning horizons,
and the need for smooth plan continuity. For the purposes of quantitative analysis, we
use the military munitions logistics planning and execution problem as a case study. The
approach taken is to break the planning horizon into readily solvable finite sections, solve
those sections using deterministic linear programming methods to produce partial plans,
and reassemble the partial plans into a continuous solution suitable for execution. The
analyses contained in this thesis focus on how the variants of this approach behave under
plausible real-world problem scenarios. In this chapter we present a summary of the
research accomplished in this thesis, and draw some conclusions from the research about
how best to apply this approach to other stochastic real-time network flow problems.
6.1 Research Summary
Chapter 1 provided the background necessary to understand flow-based networks
and minimum-cost network flow problems. We defined some important terms regarding
the different characteristics of common network flow problems, and gave examples of
several problems embodying these characteristics.
We also introduced linear
programming and mixed integer programming as techniques for solving network flow
60
problems with fixed changes or certain side constraints. The discussion then progressed
towards a definition of stochastic real-time network flow problems.
It is this type of
problem we are interested in analyzing, so we chose one - the munitions logistics
planning problem - to which to apply our approach and perform scenario analyses.
The primary objectives of the thesis were given in Chapter 2. The first was the
discovery of cost-effective decompositions of an unbounded planning horizon into
manageable partial plans under various network conditions.
The second involved
qualitatively analyzing the value of forecasting and information assessment in a real-time
flow-based network. The final objective was to measure the impact of replan delay time
on overall solution quality. Taken together, the goal of these three objectives was to
suggest a strategy for applying variants of the partial planning approach to the problem of
planning and executing solutions to real-time network flow problems.
Chapter 3 first defined the specific terms used for scenario analysis, and then
delved into the details of solving linear programs using a commercial software package.
The mechanical process of scenario analysis was also explained. Then the notion of flow
pipelining was covered.
Flow pipelining is a method for minimizing cost at the
boundaries between two consecutive partial plans.
The following chapter then explained in detail the structure of the munitions
logistics planning problem. The mechanisms of munitions containerization and ground
transportation were described and illustrated.
61
Chapter 4 also included a formalized
description of the entire problem, formulated as a mixed integer programming problem,
and provided a simplified time-space diagram of the flow network.
With the problem explained and formally presented, the next step was to carry out
scenario analyses. Each analysis, presented with results in Chapter 5, provided insight
into which variant of the partial planning approach produces the most efficient solution to
the scenario being analyzed. Individual variants included the length of each partial plan,
the frequency with which new partial plans are generated, and the length of time needed
to integrate two consecutive partial plans. Stochastic demand, transportation delays, and
the loss of network edges, sources, and sinks were all included in the analyses. The final
analysis combined these and other elements into a single overall plan execution
simulation, and measured the influence of different approach variants on the overall
solution cost.
6.2 Conclusions
A significant amount of scenario analysis has been performed to test the various
aspects of our partial planning approach. We now review these results, relate them to the
objectives stated in Chapter 2, and present several conclusions that can be drawn from
them.
The first objective was to relate the length of partial plans on a periodic
replanning schedule to overall solution efficiency under both deterministic and stochastic
62
network conditions. From the results shown in Figure 7, we can infer that longer partial
plans are monotonically better than shorter ones when demand is deterministic and
known at plan time, and no stochastic network events are present.
This result is no
particular surprise; with perfect information, linear programs are guaranteed to produce at
least as good, if not better, results when more of the problem is solved at once.
More interesting conclusions can be derived from the stochastic scenario analysis.
Longer plans on a periodic replanning schedule actually perform much worse than shorter
ones when faced with unpredictable demand. The reason is that demand not expected or
planned for is likely to remain unsatisfied for a longer period of time. Late arrivals are
severely penalized, so short partial plans that minimize loss will perform better in
execution than long partial plans that achieve greater deterministic efficiency but cannot
adequately deal with stochastic events. The relationship, however, is not quite so simple.
Very short plans also suffer from the inability to plan efficiently over a reasonable
amount of time. A very short partial plan that must deal with missed past demand as well
as demand in its own span is forced to utilize the fastest possible means of transport over
its entire span, and has no chance to use slower but cheaper means of transport before its
span ends. When planning for stochastic network demand, the best choice of plan length
falls between the two extremes. For our case study problem, that choice is approximately
40 periods per partial plan.
We turn next to the question of information forecasting. The analyses in Section
5.2 reveal that significant demand increases can have a serious negative effect on solution
63
quality if not predicted before they occur, or if the situation is not assessed and replanned
immediately after they occur. We can therefore conclude qualitatively that even one or
two period's advance notice is worth a great deal in terms of solution cost. Otherwise the
demand increase remains completely unsatisfied until the network's condition is
assessed, the event's effects are discovered, and a new partial plan is executed. This
conclusion is supported by the stochastic scenario results from Section 5.1, in which
unsatisfied demand dominated the cost of the overall solution.
The third and final objective of this thesis was to study the impact of replan delay
time on solution quality.
Recall that new partial plans generated in response to
significant network events are delayed by a certain small number of periods before being
executed, in order to approximate the blending of the old and new partial plans into a
more continuous solution.
From the results presented in Figure 13, we see that
transportation network events are easily handled even when new plans are delayed up to
four periods after the events occur. The loss of a single depot-port network link, even for
a relatively long period of time, has a relatively minor effect on the solution. We can
deduce that this result stems from the fact that, to reduce costs, the model utilizes mostly
rail traffic spread across all five depots. The loss of one only of those links does not
seriously diminish efficient flow through the network. The loss of some or the entire
truck transportation network has almost negligible effect when compared to similar rail
loss.
64
More interesting conclusions about the effects of plan continuity requirements can
be taken from the partial plan execution simulations in Section 5.5. With the inclusion of
several stochastic network events, including changes in demand, transportation delays,
and network link loss, longer replan delays have an increased negative effect over the
single event scenarios from Section 5.4. We can deduce from figure 15 that as replan
delay time increases, it has greater effect on shorter partial plans than longer ones. In
fact, the difference between 30-period partial plan solutions and 50-period partial plan
solutions is much greater than the difference between 50 and 80 period partial plan
solutions. Changes in the network that are not handled as quickly as possible, especially
when using shorter partial plans, degrade solution quality considerably. We can therefore
conclude that during real-time execution of partial planning solutions under stochastic
network conditions, implementations of plan continuity mechanisms that require more
than one or two periods to achieve are a significant detriment to overall solution
efficiency.
The final conclusions to be drawn from the research presented in this thesis are
the recommended variants of the partial planning approach to use for various types of
dynamic network flow problems. As we have learned from the analyses in Chapter 5 and
the conclusions drawn from them, the efficiency of an overall solution (given particular
values for partial plan length and partial plan integration time) is greatly affected by
network conditions. The proper choices for partial plan length and integration time must
be tailored to the network conditions of the problem being solved.
Exact values for
optimal choices can be determined only by performing scenario analyses similar to those
65
found in Chapter 5 on the specific problem, but we can infer approximate values that
should result in relatively efficient solutions. The inferred partial planning approaches
for various types of dynamic network flow problems are summarized in Figure 16.
Network Conditions
Demand Profile
Replan Schedule
Network
delay and loss
Partial Planning
Approach Variant
Yes
5 Mid-length partial plans
No
p
and short replan delay
Periodic
Very long partial plans
and any necessary delay
Deterministic
Yes
Mid-length partial plans
and short replan delay
On Deman
No
Yes
Short partial plans and
shortest possible delay
No
Mid-length partial plans
and short replan delay
Yes
Mid-length partial plans
and short replan delay
No
Long partial plans and
reasonably short delay
Periodic
Stochastic
On Demand
Figure 16: Approaches for Planning Dynamic Network Flow Problems
Sorted by network conditions. Mid-length plans are approximately 50 periods (six weeks) long.
Shorter and longer partial plans are relative to this value.
66
7. Future Work
There are three areas of future work that would further the research completed in
this thesis.
These areas are: development of better partial plan integration models,
exploration of plan correction methods, and application of the partial planning approach
to other specific stochastic real-time network flow problems. Each of these topics is
covered in more detail in the following sections.
7.1 Partial Plan Integration Techniques
The analyses performed in this thesis used delayed execution of new partial plans
generated on demand as a rough approximation of realistic partial plan integration. The
process by which an old partial plan ends and a new one begins can be quite sophisticated
for large networks. Several possible methods include beginning execution of different
parts of the new partial plan at different times, or executing individual aspects of each
(old and new) partial plan concurrently for a short time. Specific details, like the duration
of the partial plan transition period or the degree of acceptable solution discontinuity,
depend on the problem under consideration. The case study analysis in this thesis could
benefit from the implementation of a plan integration strategy that allowed each depot to
alter its transportation schedule independently of the others.
Then localized network
events, like single depot-port link loss, could be handled immediately by replanning the
67
individual depot's schedule instead of the entire network schedule.
To maintain a
globally optimal solution, the rest of the depots could replan a short time later.
Implementing more accurate partial plan integration techniques is important for
bringing planned flow execution closer to actual network operating conditions. Localized
replanning is common in real-world settings, particularly for dealing immediately with
time-critical network events like airline flight delays. Better models for such situations
should lead to more efficient and easily executable partial plans.
7.2 Plan Correction Methods
The results of the partial plan length analysis (Section 5.1) showed that longer
partial plans produce better overall solutions than shorter partial plans in deterministic
network scenarios. The long partial plans were able to optimize over a greater portion of
the overall problem horizon. But when the network included stochastic events, as is often
the case with real-time systems, the longer partial plans (on a periodic replanning
schedule) suffered from the inability to compensate for such events during execution.
Plan Correction refers to a partial plan's capacity to adapt to certain changes in network
parameters during execution and without replanning. Appropriate changes for this type
of correction include individual shipment loss and short-term single network link outages.
With a mechanism in place to correct for these occurrences without replanning, a realtime system operating in stochastic event scenarios could benefit from the reduced flow
68
cost of longer partial plans while maintaining a solution comparable to those attained by
shorter partial plans under the same conditions.
Any physical mechanism for correcting partial plans would probably have to be
implemented at the points of plan execution (i.e. the depots themselves) rather than in the
model used for planning. If such physical mechanisms could be implemented reliably
(like shipment tracking and automatic reshipment when loss occurs), then the corrective
effects of such a system could be modeled and used for planning. In Section 5.3 we
discussed a method for applying stochastic transportation delays to the planning system
by altering prior partial plan solutions. The same method can be used to account for the
corrections to these delays and other events. The result would be a planning system that
more closely matches real-world network behavior. As stated in the previous section,
more accurate planning models generally lead to better plans and solutions.
7.3 Application to Other Problems
The intent of the research presented in this thesis was to develop an approach to
solving stochastic real-time network flow problems. We have analyzed this approach
extensively against a single real-time system, munitions logistics planning and execution.
In order to determine the relevancy of the conclusions drawn in section 6.2 to other
network flow problems, they need to be tested in similar analyses of those problems.
Additional support for our conclusions (beyond the analyses in Chapter 5) can be
obtained by formulating and analyzing other stochastic real-time systems in a framework
69
similar to the one used here. This task might be made easier by the development of a
standardized analysis platform for the partial planning approach which, when given a
problem formulation and some problem-specific information, could solve a suite of
scenario analyses automatically and display relevant results.
70
Appendix
Sample AMPL Model File
# -----------------#list of named sets
# -----------------set
Ns;
#the set
of
s et Nd; #the set of
set V;
#the set of
set B;
#the set
of
depots
ports
conveyances
container types
param first
>= 0 integer;
param last
> first
integer;
set time := first..last;
#the set
of time periods
# -------- -----------------#list of network parameters
#---------- --------------param
param
param
param
param
param
param
param
param
param
param
s (B, Ns);
d (B, Nd, time);
c (B, Ns, Nd, V);
T (Ns, Nd, V);
mu (Nd, V);
n {B);
eta (Nd);
f (Nd, V};
lam (Ns);
etai (Ns);
g;
# -------------------------#list of decision variables
# -------------------------var
var
var
var
var
var
X
Q
Y
P
R
W
(b in B, i in Ns, j in Nd, k in V, t in time) >= 0 integer;
(B, Nd, V, time) >= 0 integer;
(B, Nd, V, time) >= 0 integer;
(B, Nd, time) >= 0 integer;
(B, Ns, time) >= 0 integer;
(B, Ns, time) >= 0 integer;
# ---------------------#the objective
function
# ---------------------minimize cost:
sum {t in time) ((sum (b in B, i in Ns, j in Nd, k in V) c[b,i,j,k]
X[b,i,j,k,t]) +
(sum {j in Nd, k in V) (f[j,kJ + sum (b in B) Q[b,j,k,t]))
+
g * sum (b in B) n[b] * (sum (i in Ns) R[b,i,t]
+
sum {j in Nd) (sum (k in V) Q[b,j,k,t]
+ P[b,j,t])));
-- - - - - - # ----------#list of constraint
equations
# -------------------- ---subject to supply (i in Ns, b in B):
sum ft in time) W[b,i,t] <= s[b,i];
71
*
subject to balflow fi in
W[b, i, t-1] + R[b,i,t-1]
Ns,
-
subject to balance fj in Nd,
(sum fi in Ns) (if
t-T[i,j,k]
Q[b,j,k,t] - Y[b,j,k,t
= 0;
b in B,
R[b,i,t]
t in first+l..last):
- sum {j in Nd,
k in V} X[b,i,j,k,t]
b in B, k in V, t in first+1.
>= 0 then X[b,i,j,k,t-T[i,j,k])))
subject to flowbal fj in Nd, b in B,
(sum [k in V) Y[b,j,k,t-1))
+ P[b,j,t-l]
t in
-
first+1..last):
P[b,j,t]
subject to coni (i in Ns, t in time):
sum {b in B) W[b,i,t] <= lam[i];
subject to portthru
{j in Nd, k in V,
sum Lb in B) Y[b,j,k,t] <= mu[j,k];
t
in
subject to con2 {i in Ns, t in time):
sum (b in B) n[b] * R[b,i,t]
<= etai[i];
subject to new {j in Nd, t in time):
sum (b in B) n[b] * P[b,j,t]
<= eta[j];
72
time):
= 0;
.last):
= d[b,j,t];
+ Q[b,j,k,t-1]
-
Sample AMPL Data File
#Members of named sets
# --------------------set
Ns
anad bgad taad caaa mcad;
set
Nd
motsu concord hadlock;
set
B
ci c2 c3;
set V
truck rail;
#name
#name
#name
#name
# -------------------------------#Values of all defined parameters
#and coefficients
# -------------------------------param first :
0;
param last
5;
param s (tr): cl
anad 10000 10000
bgad 10000 10000
taad 10000 10000
caaa 10000 10000
mcad 10000 10000
c2 c3
10000
10000
10000
10000
10000;
param d :=
[*,*,0] (tr): ci
motsu
0
concord
0
hadlock
0
c2
0
0
0
c3
0
0
0
(tr): ci c2 c3
0
0
0
concord
0
0
0
hadlock
0
0
0
[*, *1]
motsu
[*, *,21 (tr): ci c2 c3
motsu
0 0 0
concord
0
0
0
hadlock
0
0
0
[*, *,3] (tr): ci c2
motsu
58 56
concord
0
0
hadlock
34 38
c3
51
0
54
[*, *,4] (tr):
motsu
concord
hadlock
ci
0
23
0
c2
0
42
0
c3
0
49
0
(tr): ci
22
concord
0
hadlock
0
c2
26
0
0
c3
30
0
0;
[*,
*,5]
motsu
param c :=
[*,*,motsu, truck]:
ci
c2
c3
anad
3
3
3
[*,*motsurail]:
anad
ci
2
c2
2
c3
2
bgad
2
2
2
caaa
2
2
2
mcad
3
3
3
taad
3
3
3
bgad
1
1
1
caaa
1
1
1
mcad
2
2
2
taad
2
2
2
73
of
of
of
of
each
each
each
each
depot
port
container type
conveyance
[*,*,concord,truck]:
caaa
2
2
2
mcad
3
3
3
taad
caaa
1
1
1
mcad
2
2
2
taad :=
2
2
2
*,hadlock, truck]:
anad
bgad
caaa
ci
3
2
2
c2
3
2
2
c3
3
2
2
mcad
3
3
3
taad
3
3
3
*,hadlock,
anad
ci
2
2
c2
c3
2
mcad
2
2
2
2
2
2;
ci
c2
c3
anad
3
3
3
bgad
2
2
2
:=
3
3
3
f*,*,concord,rail]:
ci
c2
c3
anad
2
2
2
bgad
1
1
1
[*,
[*,
rail]:
bgad
1
1
1
caaa
1
1
1
param T :
[*, *, truck]:
motsu
anad
1
2
bgad
taad
1
2
caaa
1
mcad
concord hadlock
2
1
2
1
2
1
2
1
1
2
*,rail]:
motsu
3
anad
bgad
4
4
taad
caaa
4
mcad
3
concord hadlock
4
4
3
4
3
4
4
3
4
4;
taad
[*,
param mu: truck
motsu
500
concord
500
hadlock
500
rail
700
700
700;
param n := ci 100 c2 200 c3 50 c4 300 c5 100 c6 75
c7 125 c8 175 c9 200 c1O 400 c1l 100 c12 250 c13 100
c14 175 c15 200 c16 100 c17 75 c18 210 c19 50 c20 125;
param K := motsu 1000 concord 1000 hadlock 1000;
param eta := motsu 30000 concord 30000 hadlock 30000;
param f:
motsu
concord
hadlock
truck rail
4
2
4
2
4
2;
param lam := anad 600 bgad 600 taad 600 caaa 600 mcad 600;
param etai
:= anad 30000 bgad 30000
taad 30000 caaa 30000 mcad 30000;
param Ki := anad 600 bgad 600 taad 600 caaa 600 mcad 600;
param g
:
10;
74
References
1.
Ahuja, Ravindra K., Thomas L. Magnanti and James B. Orlin. Network Flows:
Theory, Algorithms, and Applications. Prentice Hall, Upper Saddle River, New
Jersey, 1993.
2. Algorithms and Theory of Computation Handbook, Mikhail J. Atallah, ed., CRC Press
LLC, 1999.
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http://kti.ms.mff.cuni.cz/-bartak/constraints/intro.htmnl [cited 27 May 1998].
4.
Coop, Andrew E. "Contingency Munitions Logistics Planning and Control: A
Framework for Analysis" CSDL-T-1308, SM Thesis in Operations Research,
Massachusetts Institute of Technology. C.S. Draper Laboratory, Cambridge, MA,
June 1998.
5.
Cormen, Thomas H., Charles E. Leiserson and Ronald L. Rivest. Introduction to
Algorithms. The Massachusetts Institute of Technology, Cambridge, MA, 1990.
6.
"Dictionary of Algorithms, Data Structures, and Problems," NationalInstitute of
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http://hissa.ncsl.nist.gov/dads/HTML/heuristic.html [cited 4 Sept. 1998].
7. "Linear Programming Frequently Asked Questions" Northwestern University and
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8. "National Transportation Statistics," Bureau of Transportation Statistics [online],
URL: http://www.bts.gov/ntda/nts/nts.html [cited 1999].
9. Nemhauser, G. L. and L. A. Wolsey. Integer and CombinatorialOptimization. Wiley
Interscience, 1988
75